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Plan: • Dispersionless Hirota type equations ` equations • Symplectic Monge-Ampere

• Integrability: the method of hydrodynamic reductions ` equations in 3D • Symplectic Monge-Ampere ` equations in 4D. Classification. Geometry • Symplectic Monge-Ampere

• Generalisations References E.V. Ferapontov, L. Hadjikos and K.R. Khusnutdinova, Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian, International Mathematics Research Notices, (2010) 496-535; arXiv: 0705.1774 (2007). ` equations, B. Doubrov and E.V. Ferapontov, On the integrability of symplectic Monge-Ampere Journal of Geometry and Physics, 60 (2010) 1604-1616. 2

Dispersionless Hirota type equations Three dimensions, u

= u(x, y, t): F (uxx , uxy , uyy , uxt , uyt , utt ) = 0

Examples (dispersionless limits of 2+1D integrable hierarchies):

1 utx − u2xx = uyy 2 uxx + uyy = eutt

dKP equation Boyer − Finley equation Higher dimensions, u

= u(x1 , ..., xn ): F (uij ) = 0

Solutions can be interpreted as Lagrangian submanifolds in the symplectic space with coordinates x1 , ..., xn , u1 , ..., un whose Gaussian image belongs to the hypersurface in the Lagrangian Grassmannian specified by F (uij ) Integrability? Classification? Geometry? 3

= 0.

` equations Symplectic Monge-Ampere = (uij ) be the Hessian matrix of a function u(x1 , ..., xn ). Symplectic ` equations are linear combinations of all possible minors of U Monge-Ampere

Let U

Examples:

First heavenly equation

u13 u24 − u14 u23 = 1

Second heavenly equation

u13 + u24 + u11 u22 − u212 = 0

Husain equation

u11 + u22 + u13 u24 − u14 u23 = 0

6D heavenly equation

u15 + u26 + u13 u24 − u14 u23 = 0

Special Lagrangian 3 − folds

Hess u = 4u

Affine spheres

Hess u = 1

Equivalence group Sp(2n) acts by linear transformations of x1 , ..., xn , u1 , ..., un Integrability? Classification? Geometry? 4

Special Lagrangian 3-folds Consider the space C3 with coordinates z 1 , z 2 , z 3 Symplectic form

(z k = xk + iuk )

ω = du1 ∧ dx1 + du2 ∧ dx2 + du3 ∧ dx3

Holomorphic volume form

Ω = dz 1 ∧ dz 2 ∧ dz 3

Im Ω = −du1 ∧du2 ∧du3 +du1 ∧dx2 ∧dx3 +dx1 ∧du2 ∧dx3 +dx1 ∧dx2 ∧du3 Special Lagrangian 3-folds are specified by the equations ω

= Im Ω = 0

In general: symplectic space R2n with coordinates x1 , ..., xn , u1 , ..., un Symplectic form

ω = du1 ∧ dx1 + ... + dun ∧ dxn

Constant coefficient differential n-form Φ in dxk , duk ` equations are specified by the equations ω Symplectic Monge-Ampere Manifestly Sp(2n) invariant

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=Φ=0

Integrability: the method of hydrodynamic reductions Applies to quasilinear systems

A(u)ux + B(u)uy + C(u)ut = 0 Consists of seeking N-phase solutions

u = u(R1 , ..., RN ) The phases Ri (x, y, t) are required to satisfy a pair of commuting equations

Ryi = µi (R)Rxi , ∂j µi Commutativity conditions: µj −µi

=

Rti = λi (R)Rxi

∂j λ i λj −λi

Definition A quasilinear system is said to be integrable if, for any number of phases N, it possesses infinitely many hydrodynamic reductions parametrised by N arbitrary functions of one variable. 6

Example of dKP 1 2 uxt − uxx = uyy 2 Quasilinear form: set v = uxx , w = uxy vt − vvx = wy , vy = wx N -phase solutions: v = v(R1 , ..., RN ), w = w(R1 , ..., RN ) where Ryi = µi (R)Rxi ,

Rti = λi (R)Rxi

Then

∂i w = µi ∂i v,

λi = v + (µi )2

Equations for v(R) and µi (R) (Gibbons-Tsarev system):

∂j v , ∂j µ = j µ − µi i

∂i v∂j v ∂i ∂j v = 2 j (µ − µi )2

In involution! General solution depends on N arbitrary functions of one variable. 7

Generalised dKP uxt − f (uxx ) = uyy Quasilinear form: set v

= uxx , w = uxy vt − f (v)vx = wy , vy = wx

N -phase solutions: v = v(R1 , ..., RN ), w = w(R1 , ..., RN ) where Ryi = µi (R)Rxi ,

Rti = λi (R)Rxi

Then

∂i w = µi ∂i v,

λi = f 0 (v) + (µi )2

Generalised Gibbons-Tsarev system:

∂j µi = f 00 (v) Involutivity

∂j v , j i µ −µ

∂i ∂j v = 2f 00 (v)

←→ f 000 = 0 8

∂i v∂j v (µj − µi )2

General case utt = f (uxx , uxy , uyy , uxt , uyt ) The integrability conditions reduce to a system of third order PDEs for f :

d3 f = S(f, df, d2 f ) In involution!

Theorem The moduli space of integrable equations of the dispersionless Hirota type is

21-dimensional. The action of the equivalence group Sp(6) on the moduli space of integrable equations possesses an open orbit

• Parametrisation by generalised hypergeometric functions (Odesskii-Sokolov) • Geometric characterisation of the associated GL(2, R) structures (Smith) 9

Further examples uxy 1 utt = + η(uxx )u2xt , uxt 6 η solves the Chazy equation η 000 + 2ηη 00 = 3(η 0 )2 . uxy + uxt uyt r(utt ) = 0, r solves the third order ODE 2

r000 (r0 − r2 ) − r002 + 4r3 r00 + 2r03 − 6r2 r0 = 0. General solution

∞ X (−1)n nq n t , q = e r(t) = 1 − 8 n 1 − q n=1

r(t) is the Eisenstein series associated with the congruence subgroup Γ0 (2) of the modular group. 10

Geometric picture

Equation F

= 0 −→ Hypersurface M 5 ⊂ Λ6

Solutions −→ Lagrangian manifolds whose Gaussian images belong to M 5

Sp(6) −→ Equivalence group of the problem Classification of equations of the dispersionless Hirota type up to Sp(6) equivalence is equivalent to the study of Sp(6) geometry of hypersurfaces

M 5 ⊂ Λ6 11

Geometry of the Lagrangian Grassmannian Action of Sp(6) on the Lagrangian Grassmannian Λ6 :

A

B

C

D

˜ = (AU + B)(CU + D)−1 ∈ Sp(6) =⇒ U ˜ ) = (...)det dU det (dU

Theorem 3 The group of conformal automorphisms of the symmetric cubic form det dU is isomorphic to Sp(6) Objects in P 5

= P T Λ6 : cubic hypersurface det dU = 0

Its singular locus is the Veronese surface V 2

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⊂ P5

Geometry of a hypersurface M 5

⊂ Λ6

The intersection of T M 5 with the Veronese surface V 2 specifies in T M 5 a rational normal curve of degree four, that is, a curve equivalent to

(1 : t : t2 : t3 : t4 ) ⊂ P 4

Thus, M 5 is supplied with a Gl(2, R)-structure

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Geometric interpretation of the integrability

1-component hydrodynamic reductions −→ secant curves 2-component hydrodynamic reductions −→ bisecant surfaces 3-component hydrodynamic reductions −→ trisecant 3-folds

Integrability ←→ existence of sufficiently many trisecant 3-folds

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` equations in 2D Symplectic Monge-Ampere Hessian matrix

U =

u11 u12

u12 u22

` equations are of the form Symplectic Monge-Ampere

M2 + M1 + M0 = 0 Explicitly,

(u11 u22 − u212 ) + au11 + bu12 + cu22 + d = 0 Any such equation is linearisable by a transformation from Sp(4)

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Geometry in 2D Symplectic space with coordinates x1 , x2 , u1 , u2 . Lagrangian planes form the Lagrangian Grassmannian Λ3

u1 u2

=

Plucker embedding of Λ3 in P 4 is (1 ¨

u11 u12

u12 u22

x1

x2

: u11 : u12 : u22 : u11 u22 − u212 )

` equations ←→ hyperplanes in P 4 Symplectic Monge-Ampere 16

` equations in 3D Symplectic Monge-Ampere Hessian matrix

u11 U = u12 u13

u13

u22

u23

u23

u33

u12

` equations are of the form Symplectic Monge-Ampere

M3 + M2 + M1 + M0 = 0 In 3D there exist three essentially different canonical forms modulo the equivalence group Sp(6) (Lychagin, Rubtsov, Chekalov, Banos):

u11 = u22 + u33 ,

Hess u = 4u,

Hess u = 1

Integrability ←→ linearisability (not true in dim 4 and higher: the first heavenly equation, u13 u24 − u14 u23 = 1, is integrable, but not linearisable). 17

Geometry in 3D Symplectic space with coordinates x1 , x2 , x3 , u1 , u2 , u3 . Lagrangian planes form the Lagrangian Grassmannian Λ6

Plucker embedding of Λ6 in P 13 ¨

` equations in 3D ←→ Integrable symplectic Monge-Ampere hyperplanes tangential to Λ6 ⊂ P 13 18

` equations in 4D Symplectic Monge-Ampere ` equation in 4D for u(x1 , x2 , x3 , x4 ) and Consider a symplectic Monge-Ampere take its traveling wave reduction to 3D,

u = u(x1 + αx4 , x2 + βx4 , x3 + γx4 ) + Q(x, x), where Q(x, x) is an arbitrary quadratic form.

Integrability in 4D ←→ linearisability of all traveling wave reductions to 3D In particular, all traveling wave reductions of the first heavenly equation,

u13 u24 − u14 u23 = 1, are linearisable.

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Classification of integrable equations in 4D linear wave

u11 − u22 − u33 − u44 = 0

first heavenly

u13 u24 − u14 u23 = 1

second heavenly

u13 + u24 + u11 u22 − u212 = 0

modified heavenly

u13 = u12 u44 − u14 u24

Husain

u11 + u22 + u13 u24 − u14 u23 = 0

general heavenly

αu12 u34 + βu13 u24 + γu14 u23 = 0, α + β + γ = 0

Conjecture In dimensions D ≥ 4, any integrable equation of the form ` type F (uij ) = 0 is necessarily of the symplectic Monge-Ampere Not true in 3D: take the dKP equation uxt

− 12 u2xx = uyy

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Geometry in 4D Symplectic space with coordinates x1 , x2 , x3 , x4 , u1 , u2 , u3 , u4 . Lagrangian planes form the Lagrangian Grassmannian Λ10

Plucker embedding of Λ10 in P 41 is covered by a 7-parameter family of Λ6 ¨

` equations ←→ hyperplanes Integrable symplectic Monge-Ampere tangential to Λ10 ⊂ P 41 along a four-dimensional subvariety X 4 which meets all Λ6 ⊂ P 41 21

Generalisations Gr(d, n): Grassmannian of d-dimensional subspaces of a linear space V n . X ⊂ Gr(d, n): submanifold of the Grassmannian, dim X = n − d. Σ(X): differential system governing d-dimensional submanifolds of V n whose Gaussian image is contained in X . Conjectures

• For d ≥ 3, n − d ≥ 2 the moduli space of integrable systems Σ(X) is finite dimensional

• For d ≥ 4, n − d ≥ 2 any integrable system Σ(X) must be linearly ` property: systems of degenerate (generalisation of the Monge-Ampere ` type correspond to linear sections of the Plucker Monge-Ampere ¨ embedding of Gr(d, n)).

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